We have already seen different simple interpolation methods that take space into account when estimating unknown values, mainly by incorporating the distance between locations of measured values and the prediction location.
Next, we will look at geostatistic interpolation methods that also take into account knowledge about how certain variables are spatially (auto)correlated.
First, let´s recap some basic concepts of correlation.
Random variables (RVs) are numeric variables whose outcomes are subject to chance.
The cumulative distribution of probability \(F_x(\cdot)\) over outcomes \(z\) over all possible values of the RV \(Z\) is the probability distribution function:
\[P(Z \le z) = F_Z(z) = \int_{-\infty}^z f_Z(u)du\] where \(f_Z(\cdot)\) is the probability density function of \(Z\). The sum of all probability is 1.
Try to think of \(E(Z)\) as \(\frac{1}{n}\sum_{i=1}^{n} z_i\), with \(i \rightarrow \infty\).
Two random variables \(X\) and \(Y\) have covariance defined as \(\Cov(X,Y) = E[(X-E(X))(Y-E(Y))]\)
Correlation and covariance
Correlation is scaled covariance, scaled by the variances. For two variables \(X\) and \(Y\), it is \[\Cor(X,Y) = \frac{\Cov(X,Y)}{\sqrt{\Var(X)\Var(Y)}}\]
it is quite easy to show that \(|\Cov(X,Y)| \le \sqrt{\Var(X)\Var(Y)}\), so correlation ranges from -1 to 1
for this, note that \(\Cov(X,X)=\Var(X)\) and \(\Cov(X,-X)=-\Var(X)\).
It is perhaps easier to think of covariance as unscaled correlation.
Note: A large covariance does not imply a strong correlation
Random variable:\(Z\) follows a probability distribution, specified by a density function \(f(z)= \Pr(Z=z)\) or a distribution function \(F(z)=\Pr(Z \le z)\)
Expectation:\(\E(Z) = \int_{-\infty}^{\infty} f(s)ds\) – center of mass, mean.
Variance:\(\Var(Z)=\E(Z-\E(Z))^2\) – mean squared distance from mean; measure of spread; square root: standard deviation of \(Z\).
Covariance:\(\Cov(X,Y)=\E((X-\E(X))(Y-\E(Y)))\) – mean product; can be negative; \(\Cov(X,X)=\Var(X)\).
Correlation:\(r_{XY}=\frac{\Cov(X,Y)}{\sqrt{\Var(X)\Var(Y)}}\) – normalized \([-1,1]\) covariance. -1 or +1: perfect correlation.
Correlation
What is spatial correlation?
Waldo Tobler’s first law in geography:
“Everything is related to everything else, but near things are more related than distant things.” [Tobler, 1970, p.236]*
But how then is “being related” expressed?
TOBLER, W. R. (1970). “A computer model simulation of urban growth in the Detroit region”. Economic Geography, 46(2): 234-240.
What is spatial correlation?
Spatial correlation can be explored in different ways.
One way is to take up an idea from time series: look at lagged correlations, and the \(h\)-scatterplot.
What is it? Plots of (or correlation between) \(Z(s)\) and \(Z(s+h)\), where \(s+h\) is \(s\), shifted by \(h\) (time distance, spatial distance).
In geostatistics the spatial correlation is modelled by the semivariogram instead of a covariogram (or correlogram). The term variogram is used synonymously with semivariogram. The (semi) variogram plots semivariance as a function of distance.
\(C(h)\) needs (an estimate of) \(m\), \(\gamma(h)\) does not
\(C(0)\) may not exist (\(\infty\)!), when \(\gamma(h)\) does (e.g., Brownian motion)
Anisotropy
Some processes are directionally dependent (anisotropic), i.e. do not have identical properties in all directions. When investigating such phenomena the semivariance does not only depend on the distance between two points but also on the direction of the distance vector.
example: global annual mean temperature.
Isotropic (left) vs anisotropic (right) process
Check for Isotropy/Anisotropy
group values not only regarding distance but also direction of the distance vector
In order to be able to estimate spatial correlation from observational data, we need to assume intrinsic stationarity.
This assumes the underlying process to be a random function composed of a mean and residual
\(Z(s) = m + e(s)\)
with a constant mean
\(E(Z(s)) = m\)
and a variogram defined as
\(\gamma(h)= \frac{1}{2}E(Z(s)-Z(s+h))^2\)
This imlplies that the variance of \(Z\) is constant, and the spatial correlation of \(Z\) does not depend on location \((s)\), but only on separation distance \((h)\).
Simulation examples
(Co)Variograms and random fields
Given a theoretical (co)variogram, we can create processes (random fields) that have the desired properties.
In the following, we create different example simulations that show for an (artificial) variable how different variogram properties are associated with different spatial distributions of the values of that variable.
